Session B.pdf - Clarkson University

If an ice cover presents, the composite resistances are respectively:
2 2 2
2 2 2
ρ g n
C
u + v
ρ g n
C
u + v
τ
i x
+ τb
x
=
u τ v
1/ 3
i y
+ τb
y
=
, (5)
1/ 3
h
h
where, n b =bed roughness; n c =composite Manning’s roughness. In this paper, the
composite Manning’s coefficent is taken to be (Mao and Ma, 2002)
⎛ P n +
⎜
⎝ P
P
n
⎞
⎟
⎠
3/2
3/2
b b i i 2/3
n = ⎜
⎟
C
. (6)
Water temperature can be expressed as a two-dimensional convection-diffusion equation
in terms of conservation of thermal energy:
∂
∂t
( ρC
AT) + ( QρC
T) + ( QρC
T)
P
∂ ⎛
= ⎜AE
xρC
∂x
⎝
P
∂
∂x
∂T
⎞
∂x
⎟ +
⎠
P
∂
∂y
∂ ⎛
⎜AE
yρC
∂y
⎝
P
P
=
∂T
⎞
⎟ + B Σ S,
∂y
⎠
where T = water temperature; C p = specific heat of water; A = area; B = surface width;
E x , E y are dispersion coefficient in x- and y-direction respectively; ∑S = net heat flux
from water to air, per unit surface area of flow. Assuming E x = E y = E and Q remaining
constant, the above equation can be re-written as
2
∂ T ∂ T ∂ T ⎛ ∂ T
+ u + v = E
2
t x y
⎜
∂ ∂ ∂ ⎝ ∂ x
2
∂ T ⎞
+ +
2
y
⎟
∂ ⎠
ρ
(7)
B
Σ S . (8)
C A
The equations for surface and suspended ice are respectively (Lal and Shen, 1991)
P
∂ C
∂ t
S
∂ C
+ u
∂ x
S
∂ C
+ v
∂ y
S
=
−
ρ
i
B
L
i
A
∑ S
+
α
A
⎛ V
⎜ −
⎝
⎞
⎟
⎠
C
z
1
C
u ⎟ ; (9)
i
∂ C
∂ t
c
∂ CC
+ u
∂ x
∂ CC
+ v
∂ y
= −
ρ
i
B
L
i
∑ S
A
V
−
α A
⎜
⎛ −
⎝
⎞
⎟
⎠
z
1 CC
u ⎟ , (10)
i
where, C s = surface-ice volumetric concentration; C c = suspended ice volumetric
concentration; L i = latent heat of water fusion; α = an empirical coefficient quantifying
the rate of supply to the surface ice from suspended frazil ice (Wu, 2002) ; u i = buoyant
velocity; V z = vertical component of current turbulence. From the mass conservation of
surface ice at the leading edge, the rate of ice-cover progression of leading edge is (Lal
and Shen, 1991):
V
P
=
B 0
t
i
Q
i
S
−
Q
i
( 1 − e
j
) − ( QS
− Qu
)/VSCP
u
, (11)
where V p = progression rate; e j = overall porosity of ice cover; Q u = volumetric rate of
ice swept at the leading edge;
i
Q
S = volumetric rate of surface-ice discharge; B 0 = top
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